How to define uniformly ticking clock defined in GR?

[center o bass]

I just read the following definition of a clock in GR:

"A clock is a smooth embedding γ : t → γ(t) from a real interval into M such that the tangent vector \dot{γ} (t) is everywhere timelike with respect to g and future-pointing. This terminology is justified because we can interpret the value of the parameter t as the reading of a clock. Note that our definition of a clock does not demand that “its ticking be uniform” in any sense. Only smoothness and monotonicity are required."

And I wondered how would a clock that is "ticking uniformly" be defined?

 

[UltrafastPED]

Since the spacetime intervals are relativistic invariants, and on a time-like path they are non-zero - you could start by trying uniform spacetime intervals and see how they do.

At least everybody would agree with the ticks ...

 

[DrGreg]

Using that notation, "ticking uniformly" would mean the length of the tangent vector $\dot{γ} (t)$ was constant.

The length of the tangent vector $\dot{γ} (t)$ is proportional to the ratio of proper time to clock time $\|\dot{γ} (t)\| = c\,d\tau/dt$.